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Theorem cotr3 14326
Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr3.a 𝐴 = dom 𝑅
cotr3.b 𝐵 = (𝐴𝐶)
cotr3.c 𝐶 = ran 𝑅
Assertion
Ref Expression
cotr3 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3
StepHypRef Expression
1 cotr3.a . . 3 𝐴 = dom 𝑅
21eqimss2i 4023 . 2 dom 𝑅𝐴
3 cotr3.b . . . 4 𝐵 = (𝐴𝐶)
4 cotr3.c . . . . 5 𝐶 = ran 𝑅
51, 4ineq12i 4184 . . . 4 (𝐴𝐶) = (dom 𝑅 ∩ ran 𝑅)
63, 5eqtri 2841 . . 3 𝐵 = (dom 𝑅 ∩ ran 𝑅)
76eqimss2i 4023 . 2 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
84eqimss2i 4023 . 2 ran 𝑅𝐶
92, 7, 8cotr2 14325 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wral 3135  cin 3932  wss 3933   class class class wbr 5057  dom cdm 5548  ran crn 5549  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559
This theorem is referenced by: (None)
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